Nonpermuting limits in the theory of radiating electrons: Zero damping versus infinite volume

The quantum mechanics of a harmonically bound electron with standard dipole model coupling to the electromagnetic field is studied exactly in the limit of zero damping (charge e→0) and infinite field extension (R→∞). In the limit sequence (i) R→∞, (ii) e→0, one obtains Gaussian wave functions for all stationary states (n = 0, 1, 2, …) of the electron. These Gaussians do have the usual variances 〈x2μ〉n (μ = 1, 2, 3) but otherwise obviously differ from the standard free harmonic oscillator result. The latter arises only in the limit sequence (i) e→0, (ii) R→∞.